To find the quadratic function representing the total area of the park and the surrounding path, follow these steps:
1. Define the dimensions of the park:
- Let [tex]\( x \)[/tex] be the width of the park.
- The length of the park is given to be three times its width, so the length is [tex]\( 3x \)[/tex].
2. Consider the path surrounding the park:
- The path adds 7 feet to each side of the park's dimensions.
- Therefore, the width including the path is [tex]\( x + 2 \times 7 = x + 14 \)[/tex].
- Similarly, the length including the path is [tex]\( 3x + 2 \times 7 = 3x + 14 \)[/tex].
3. Calculate the total area of the park and the path:
- The area can be found by multiplying the modified dimensions:
[tex]\[
\text{Total Area} = (x + 14)(3x + 14)
\][/tex]
4. Expand and simplify the expression to obtain the quadratic function:
- Use the distributive property (FOIL method) to expand the terms:
[tex]\[
(x + 14)(3x + 14) = x \cdot 3x + x \cdot 14 + 14 \cdot 3x + 14 \cdot 14
\][/tex]
- Perform the multiplications:
[tex]\[
= 3x^2 + 14x + 42x + 196
\][/tex]
- Combine like terms:
[tex]\[
= 3x^2 + 56x + 196
\][/tex]
Therefore, the quadratic function representing the total area of the park and the surrounding path is:
[tex]\[
3x^2 + 56x + 196
\][/tex]
